Cross product: Proof determinant equals area of parallelogram

In the cross product u x v, if it's defined as a determinant, then it is not obvious why the length of u x v should equal the area of the parallelogram between them. This video presents a clever but simple proof.

[josephwallis]

Good decent mathematical work, thank you. This benefits those high school exam students who want to get top distinction marks. I know it bc I came third in high school cert public highest maths in Australia. If you find that the cross product's magnitude is already defined as determinant. This is not so much so a proof but to show case the logic of it. There must be some exceptions if mathematicians prefer to define it. Mathematicians are people I admire a lot. Kudos

[zak]

Amazing video! Really easy to understand and is just really well put together!

[yordangrigorov]

Thank you for the video! But I came here to learn precisely about the step that you decided to miss. May I suggest a follow up video? You rock!

[kmehour]

Greate Video, but I didn't quite get the missing logical step, would someone care to explain ?

[KavinduFernando-pe]

grate video, really help fulll, if its possible can u explain why UxV = det(that 3 by 3 matrix)

[VeerapatWongsirojkul-sr]

Thank you so much!!!, I had been thinking for so long how does this make sense. Your video help me a lot ^_^

[abhikigaming]

Bro just verified the formula and called it a proof.

[borissimovic]

Can someone please explain, if you can how the length of a vector, which is 1D “a line” (length of cross product vector) can be equal to the area which is 2D (2d space includes more information width and length) for example something has a length of 10cm, not 10cm2, how can length be expressed in area terms since the length is 1D and area is 2D what is the intuition, can someone explain?

[fabiojr]

Where does the sine comes in the formula?